Consider a function $f:[0,1]\to[0,1]$. Suppose that there is a real number $0<\lambda<1$ such that $f(\lambda)\notin\{0,\lambda\}$ and the equality $$f(f(x)+y)=f(x)+f(y)$$holds whenever the function is defined on the arguments. (a) Give an example of such a function. (b) Prove that for some $x\in[0,1]$, $$\underbrace{f(f(\cdots f(}_{19}x)\cdots))=\underbrace{f(f(\cdots f(}_{98}x)\cdots)).$$
Problem
Source: Ukraine 1998 Grade 11 P4
Tags: fe, functional equation, Functional Equations, algebra